The Riemann Hypothesis (RH) has a strong claim to being the outstanding open
problem in mathematics. Much has been written about RH, but rarely with
anything like the scope that Conrey covers in but a dozen pages
(The
Riemann Hypothesis, Notices Amer. Math. Soc. 50 (2003), 341-353), outlining the
mathematical context that justifies the importance of RH, key moments in the
problem's 140 plus-year history, known partial results and blind alleys, various
threads of numerical and theoretical evidence, and suggestive connections with
disparate branches of mathematics and theoretical physics. The mathematical
exposition is enhanced by the judicious use of anecdotes illustrating the human
drama of the quest for a proof and of figures that help the reader visualize
the zeta function as a function of a complex variable and the key connections
between the distribution of prime numbers, the distribution of the zeros of the
Riemann zeta function, and conjecturally also the distribution of the eigenvalues
of random Hermitian operators.

Conrey remarks on one of those fascinating connections (Gauss's class number
problem and a "conspiracy of L-functions") that "we seem to be players in the
middle of a mystery novel." The same can be said of the status of the Riemann
Hypothesis itself. Conrey has given a masterly and lucid introduction to the plot
thus far, to the detectives who brought us to this point, and to what may be called
the main suspects: the mathematical structures that might be expected to figure
in the eventual resolution of this central mystery of modern mathematics.